1 Unless otherwise noted, the content of this course material is licensed under a Creative Commons Attribution – Non-Commercial 3.0 License. Copyright 2008, Yan Chen You assume all responsibility for use and potential liability associated with any use of the material. Material contains copyrighted content, used in accordance with U.S. law. Copyright holders of content included in this material should contact with any questions, corrections, or clarifications regarding the use of content. The Regents of the University of Michigan do not license the use of third party content posted to this site unless such a license is specifically granted in connection with particular content objects. Users of content are responsible for their compliance with applicable law. Mention of specific products in this recording solely represents the opinion of the speaker and does not represent an endorsement by the University of Michigan. For more information about how to cite these materials visit

2 Dominance and Nash Equilibrium SI 563 Lecture 2 Professor Yan Chen Fall 2008 Professor Yan Chen Fall 2008 Some material in this lecture drawn from

3 Agenda – Dominance and best response »Dominance »Best response »Dominant strategy equilibrium –Rationalizability and iterated dominance »Dominance-solvable equilibrium –Nash equilibrium »Pure strategy Nash equilibrium »Mixed strategy Nash equilibrium

4 Dominance and Best Response (Watson Chapter 6)

5 Example: Prisoners’ Dilemma ConfessNot Confess Confess -5, -50, -15 Not Confess -15, 0-1, -1 Tchaikovsky Conductor

6 Solving a strategic form game: Best response (reply) A strategy is a best response (reply) to a particular strategy of another player, if it gives the highest payoff against that particular strategy How to find best responses –Discrete strategy space: for each of opponent’s strategy, find strategy yielding best payoff –Continuous strategy space: use calculus

7 Best Response: Prisoners’ Dilemma ConfessNot Confess Confess -5, -50, -15 Not Confess -15, 0-1, -1 Tchaikovsky Conductor

8 Best response: Battle of Sexes 1, 3 0, 0 3, 1 A 2 1 A B B

9 Solving strategic form games: Dominance Confess is a best reply regardless of what the other player chooses Strategy s 1 strictly dominates another strategy s 2, if the payoff to s 1 is strictly greater than the payoff to s 2, regardless of which strategy is chosen by the other player(s). Or u i (s 1, t) > u i (s 2, t), for all t. Strategy s 2 : strictly dominated strategy

10 Examples of dominance: 2, 3 5, 0 1, 0 4, 3 U 2 1 L D R For player 1, U strictly dominates D.

11 Weak Dominance Strategy s 1 weakly dominates another strategy s 2, if the payoff to s 1 is at least as good as the payoff to s 2, regardless of which strategy is chosen by the other player(s). Or u i (s 1, t) ≥ u i (s 2, t) for all t, and u i (s 1, t’) > u i (s 2, t’), for some t’ In this case, strategy s 2 is called a weakly dominated strategy.

12 Example of strict and weak dominance: 8, 30, 44, 4 4, 21, 55, 3 3, 70, 12, 0 U M D LC 2 1 R For player 1, M strictly dominates D, U weakly dominates D. Player 2: C weakly dominates R.

13 Example of dominance: 4, 10, 2 0, 04, 0 1, 31, 2 U M D LR 2 1 Randomize between U and M dominates D, or D is dominated by the mixed strategy (½, ½, 0). Randomize between U and M dominates D, or D is dominated by the mixed strategy (½, ½, 0).

14 Dominant strategy equilibrium If every player has a dominant strategy, the game has a dominant strategy equilibrium (solution). Dominant strategy axiom: if a player has a dominant strategy, she will use it. Problem with dominant strategy equilibrium: in many games there does not exist one

15 DSE: Prisoners’ Dilemma ConfessNot Confess Confess -5, -50, -15 Not Confess -15, 0-1, -1 Tchaikovsky Conductor (Confess, Confess) is a dominant strategy equilibrium.

16 Is this a good outcome? So, both confess Question: They both would be better off with (Not, Not). So why don’t they play “Not Confess”? Tchaikovsky Conductor ConfessNot Confess Confess-5, -50, -15 Not Confess -15, 0-1, -1

17 Efficiency and equilibrium Game equilibrium is a characterization of the outcome of individually rational behavior –Because of strategic interactions, rational behavior does not always lead to outcomes that are mutually the best Dominant strategy equilibrium in Prisoner’s Dilemma: (Confess, Confess) But this is not socially efficient: both players are better off with (Not Confess, Not Confess) Many applications –Arms race –Tragedy of commons

18 Pareto Optimality A solution is Pareto optimal if and only if there is no other solution that is (1) Better for at least one agent (2) No worse for everyone else A mild (weak) criterion for social efficiency The Prisoner’s Dilemma solution is not Pareto optimal

19 Example: (Low, Low) is DSE Low priceHigh price Low price0, 00, 050, -10 High price-10, 5010, 10 Firm B Firm A

20 What to do when equilibrium is inefficient? Can’t always be improved (arms race not an easy problem!) Opportunities: –Collude / cooperate (sometimes illegal!) »OPEC »marriage »Might involve side payments if not win-win –Design systems to increase trust –Repeated interactions »Build trust »Or create opportunities for punishment!

21 Rationalizability and Iterated Dominance (Watson Chapter 7)

22 Dominance Solvability In some games, there might not be a dominant strategy, but there are dominated strategies (i.e., bad) If we can reach a unique strategy vector by iterated elimination of dominated strategies, the game is said to be dominance solvable.

23 Example: Playing mind games A B XY 2 1 3,30,50,4 0,03,11,2 Z If you are player 1, which strategy should you play?

24 FIGURE 7.2 (a) Iterative removal of strictly dominated strategies. A B XY 2 1 3,30,50,4 0,03,11,2 Z

25 FIGURE 7.2 (b) Iterative removal of strictly dominated strategies. A B XY 2 1 3,30,50,4 0,03,11,2 Z

26 FIGURE 7.2 (c) Iterative removal of strictly dominated strategies. A B XY 2 1 3,30,50,4 0,03,11,2 Z

27 Rationalizable Strategies The set of strategies that survive iterated dominance is called the rationalizable strategies Logic of rationalizability depends on –Common knowledge of rationality –Common knowledge of the game

28 Example: rationalizability/iterated dominance 5,10,41,0 3,10,03,5 3,34,42,5 U M D LC 2 1 R L is strictly dominated by (0, ½, ½), etc. Set of rationalizable strategies is {(M, R)}.

29 Strategic Uncertainty Rationalizability requires players’ beliefs and behavior be consistent with common knowledge of rationality It does not require that their beliefs be correct It does not help solve the strategic uncertainty in coordination games

30 Coordination game: want to go to an event together, with slightly different preferences Any dominant strategies? Any dominated strategies? 2, 1 0, 0 1, 2 Opera Movie OperaMovie 2 1 Strategic Uncertainty: Battle of the Sexes

31 Example: Stag hunt 5,50,4 4,04,4 Stag 2 1 Hare Any dominant strategies? Any dominated strategies? Pareto optimal outcomes?

32 Facilitate Coordination Focal point –Schelling : The strategy of conflict –Rome Institutions, rules, norms Communication

33 Nash Equilibrium (Watson Chapters 9, 11)

34 Pure Strategy Nash Equilibrium A set of strategies forms a Nash equilibrium if the strategies are best replies to each other Recall: A strategy is a best reply to a particular strategy of another player, if it gives the highest payoff against that particular strategy

35Hawk-Dove In this situation, the players can either choose aggressive (hawk) or accommodating strategies From each player’s perspective, preferences can be ordered from best to worst: –Hawk – Dove –Dove – Dove –Dove – Hawk –Hawk – Hawk The argument here is that two aggressive players wipe out all surplus

36 Hawk-Dove Analysis We can draw the game table as: Best Responses: –Reply Dove to Hawk –Reply Hawk to Dove Equilibrium –There are two equilibria –(Hawk, Dove) –(Dove, Hawk) HawkDove Hawk0, 04, 1 Dove1, 42, 2

37 FIGURE 9.2 (1) Equilibrium and rationalizability in the classic normal forms 1,-1-1,1 1,-1 H 2 1 H T T Matching Pennies

38 FIGURE 9.2 (2) Equilibrium and rationalizability in the classic normal forms 2,20,3 3,01,11,1 Not Confess 2 1 Confess Prisoners’ Dilemma

39 FIGURE 9.2 (3) Equilibrium and rationalizability in the classic normal forms 2,12,10,0 1,21,2 Opera 2 1 Movie Battle of the Sexes

40 FIGURE 9.2 (4) Equilibrium and rationalizability in the classic normal forms 0,03,13,1 1,31,32,2 H 2 1 H D D Hawk-Dove/Chicken

41 FIGURE 9.2 (5) Equilibrium and rationalizability in the classic normal forms 1,11,10,0 1,11,1 A 2 1 A B B Coordination

42 FIGURE 9.2 (6) Equilibrium and rationalizability in the classic normal forms 2,22,20,0 1,11,1 A 2 1 A B B Pareto Coordination

43 FIGURE 9.2 (7) Equilibrium and rationalizability in the classic normal forms 4,22,32,3 6,-10,0 P S D P D D Pigs

44 FIGURE 9.3 (a) Determining Nash equilibria. J 2 1 X K Y 5,63,70,4 8,33,15,2 7,54,45,6 3,57,53,3 Z L M

45 FIGURE 9.3 (b) Determining Nash equilibria. J 2 1 X K Y 5,63,70,4 8,33,15,2 7,54,45,6 3,57,53,3 Z L M

46 Mixed Strategies (Watson Chapter 11)

47 Tennis Anyone? Server Receiver Source: John Morgan, gametheory.net

48 Serving S R Source: John Morgan, gametheory.net

49 Serving S R Source: John Morgan, gametheory.net

50 The Game of Tennis Server chooses to serve either left or right Receiver defends either left or right Better chance to get a good return if you defend in the area the server is serving to Source: John Morgan, gametheory.net

51 Game Table Receiver Server LeftRight Left ¼, ¾¾, ¼ Right ¾, ¼ ¼, ¾ Source: John Morgan, gametheory.net

52 Game Table Receiver Server LeftRight Left¼, ¾¾, ¼ Right¾, ¼¼, ¾ For server: Best response to defend left is to serve right Best response to defend right is to serve left For receiver:Just the opposite Source: John Morgan, gametheory.net

53 Nash Equilibrium Notice that there are no mutual best responses in this game. This means there are no Nash equilibria in pure strategies But games like this always have at least one Nash equilibrium What are we missing? Source: John Morgan, gametheory.net

54 Extended Game Suppose we allow each player to choose randomizing strategies For example, the server might serve left half the time and right half the time. In general, suppose the server serves left a fraction p of the time What is the receiver’s best response? Source: John Morgan, gametheory.net

55 Calculating Best Responses Clearly if p = 1, then the receiver should defend to the left If p = 0, the receiver should defend to the right. The expected payoff to the receiver is: »p ¾ + (1 – p) ¼ if defending left »p ¼ + (1 – p) ¾ if defending right Therefore, she should defend left if »p ¾ + (1 – p) ¼ > p ¼ + (1 – p) ¾ Source: John Morgan, gametheory.net

56 When to Defend Left We said to defend left whenever: »p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾ Rewriting »p > 1 – p Or »p > ½ Source: John Morgan, gametheory.net

57 Receiver’s Best Response p Left Right ½ Source: John Morgan, gametheory.net

58 Server’s Best Response Suppose that the receiver goes left with probability q. Clearly, if q = 1, the server should serve right If q = 0, the server should serve left. More generally, serve left if »¼ q + ¾ (1 – q) > ¾ q + ¼ (1 – q) Simplifying, he should serve left if »q < ½ Source: John Morgan, gametheory.net

59 Server’s Best Response LeftRight ½ q Source: John Morgan, gametheory.net

60 Putting Things Together ½ q p Server’s best response 1/2 Receiver’s best response Source: John Morgan, gametheory.net

61 Equilibrium ½ q p Server’s best response 1/2 Receiver’s best response Mutual best responses Source: John Morgan, gametheory.net

62 Mixed Strategy Equilibrium A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses In the tennis example, this occurred when each player chose a mixture of left and right. Source: John Morgan, gametheory.net

63 General Properties of Mixed Strategy Equilibria A player chooses his strategy so as to make his rival indifferent A player earns the same expected payoff for each pure strategy chosen with positive probability Funny property: When a player’s own payoff from a pure strategy goes up (or down), his mixture does not change Source: John Morgan, gametheory.net

64 Does Game Theory Work? Walker and Wooders (2002) –Ten grand slam tennis finals –Coded serves as left or right –Determined who won each point Tests: –Equal probability of winning »Pass –Serial independence of choices »Fail Source: John Morgan, gametheory.net

65 Find all NE: Battle of the Sexes Chris Pat OperaBoxing Opera3,10,0 Boxing0,01,3 Source: John Morgan, gametheory.net

66 Find all NE: Hawk-Dove Krushchev Kennedy HawkDove Hawk0, 04, 1 Dove1, 42, 2 Source: John Morgan, gametheory.net

67 Highlights Dominance Rationalizability and iterated dominance Nash equilibrium –Pure strategy NE –Mixed strategy NE

68 Homework Assignment Chapter 6: #1 Chapter 7: #1, 2, 3 Chapter 11: #4, 6